Although Eq. (3.88) cannot generally be inverted with the current mathematical technology, if we restrict to points close to the boundary it turns out that it is possible to reconstruct bulk information from the boundary. In this case, the object recovered is the bulk stress-energy tensor. As in Sec. 3.6.2, we restrict our analysis to small radiiEdRd1, and to bulk fields dual to operators with scaling dimension d/2<∆< d.19
We start from Eq. (3.65),
∂R2 +R−1∂R−R−2
S(ρ|ρvac) = 16π2GN Z
∂M2
p|g|E. (3.89)
When the background is empty AdS, the minimal surface∂M2 ending on a sphere is hyperbolic, and so it is totally geodesic (meaning that all geodesics on the minimal surface are also geodesics on the constant time slice in which the minimal surface is embedded). When ∂M2 is totally geodesic, the right-hand side of Eq. (3.89) is the Radon transform, and its inverse for hyperbolic spaces exists in the mathematics literature [55,56], as we now explain.
18A related subtlety, as explained by Iyer and Wald [44], is that adding toB any function only dependent on the intrinsic geometry on∂Mdoes not change Eq. (3.87). OnB, this gauge freedom is fixed by the requirement to recover the modular Hamiltonian expectation value, however there is no similar requirement on ˜B. Any inverse Radon transform reconstruction, if it exists, would have to somehow perform this gauge fixing on ˜B.
19It it not hard to see that in this limit we should recover the boundary stress-energy tensor, and not the action: Eq. (3.88) holds when the right-hand side is purely geometric (and off-shell), in which case taking the variation ∆ turns the integrand into the geometric part of the equations of motion, which equals the bulk stress-energy tensor. This is another way of recovering Eq. (3.65).
For aD-dimensional space and a function f, the Radon transformRf integrates f on a surface that is a totally geodesic submanifold of dimension n < D, and associates the result to the surface in the space of totally geodesic submanifolds.
The inverse Radon transform R∗Rf works backwards: It integrates over totally geodesic submanifolds and associates the result to a point in the usual space; the result is just the value of the function at that point.
For odd d and totally geodesic submanifolds of dimension n = d−1, according to [55], the (inverse and direct) Radon transforms obey the identity
f = 1
(−4)(d−1)/2πd/2−1Γ(d/2)Q(∆)R∗Rf, (3.90) withf a test function (defined on the usual space) andQ(∆) a polynomial built out of the Laplace-Beltrami operator∆,
Q(∆) = [∆+ 1·(d−2)] [∆+ 2·(d−3)]·. . .·[∆+ (d−2)·1].
Applying Eq. (3.90) to (3.89), we thus obtain the energy density at a point in the bulk in terms of the boundary relative entropy as
E= 1
(−4)(d+3)/2πd/2+1Γ(d/2)GN ×Q(∆)R∗ ∂R2 +R−1∂R−R−2
S(ρ|ρvac). (3.91)
Eq. (3.91) is a toy (and yet quite complicated) example of bulk data reconstruc- tion in terms of boundary information. There exists a similar formula for even d, see [56].
At this point we should remember that Eq. (3.91) is approximate, and the ap- proximations creep in several places: (1) the formula we inverted, Eq. (3.89), is approximate and only valid near the boundary, (2) the inversion formula (3.90) is
valid for hyperbolic spaces (with no backreaction), and (3) there are totally geodesic surfaces that pass through the reconstruction point at z and go deep into the bulk (but their contributions are negligible whenEz 1, with E the typical energy scale of the CFT). Thus, it would be interesting to obtain an exact inversion formula of Eq. (3.88).
Chapter 4 Conclusion
4.1 Future directions
We now discuss some of the natural follow-up directions that arise from the work in this thesis. Some of these ideas may not be too hard to put in practice, while others may be considerably more difficult.
• A straightforward (if not too exciting) direction would be to give examples of some theories that satisfy the inequality constraints derived in this thesis, and examples that don’t. Apart from answering a very natural question, this would potentially shed light on what goes wrong when our constraints are not satisfied. It is tempting to speculate that the pathology has to do with too much negative energy density in the bulk, and consequently with a Hamiltonian becoming unbounded from below, but it would be desirable to make this precise.
• Another open thread is generalizing our discussion to time-dependent cases.
Although most of the thesis can be extended to discussions on maximin surfaces (even if we did not spell this out), Sec. 3.7 on holographic reconstruction will need considerably more work. However, the computations needed are similar
in spirit to the computations appearing in the HRT proposal, so there is some hope this could be made to work.
If the discussion is expanded to the time-dependent case, it will probably in- crease the discerning power of the constraints, since now they will also explicitly know about the dynamics of the theory.
• Obtaining the boundary dual of the Einstein equations in the nonlinear regime.
Although this seems like a most natural question to ask, it may turn out to not be too closely related to the discussion in this thesis, and the linearized result may turn out to have been some sort of coincidence. This is because answering it will almost certainly require new ingredients on the CFT side, which reduce to the first law of entanglement entropy in the linearized regime, and it is not clear what these ingredients should be. A (possibly related) complication is the existence of entanglement shadows: bulk regions not probed by any minimal surface, which exist even in nice geometries (such as AdS stars) [57]. Since the Einstein equations hold everywhere, it is tempting to conjecture they should not be associated to minimal surfaces.
• Connecting our results to the Wang-Yau quasilocal energy and mass. This is a very exciting direction, but it may be hard to put in practice. Our results are (superficially at least) similar to the features of the Wang-Yau quasilocal mass, and it is tempting to conjecture they may be related. However, the definition of the Wang-Yau quasilocal mass is very different from our our definition of quasilocal energy, and it is not clear the two can be reconciled. It would be very interesting to do so, or to prove that no reconciliation exists. A related question is whether our constraints are implied by bulk energy conditions, or whether they imply some energy condition.
• Radon transform inversion for asymptotically AdS spaces. This is almost cer- tainly very hard, but it may not be impossible, at least in certain cases. Since the mathematics literature on the subject is currently lacking, some radical ap- proaches would be needed. Two ideas come to mind: (1) Based on expression (3.90), make some informed guesses as to what the inversion formula should be in general, and try to check these guesses against some examples, deferring proof for later, and (2) Try to use advanced number-theoretic machinery, using as inspiration what was done in [58]. The first approach has the disadvantage that it is accidental, so even if it works, it will not give immediate understand- ing, and the second approach has the disadvantage that it will probably be of considerable difficulty to set up the necessary machinery, if it even is possible.
However, given sufficient time, if this second approach works it will provide a beautiful connection between general relativity, integral geometry and number theory.
• Is there a relation between our work and the “Complexity Equals Action”
story [59,60]? The reason to suspect this is that the bulk action plus boundary term L−dB plays a central role in both stories. However, the ways L−dB enters are very different, and a priori there is no motivation to connect the two, since [59,60] use the action in conjunction with non-minimal extremal surfaces, for which our results have no predictive power. But it may not be too far-fetched to speculate that with one more conceptual leap, a natural connection could be made.